We start from two charged spherical balls each of radius R with equal and opposite charge densities +ρ and -ρ. The centre of the balls are at respectively so the equation of their surfaces are
considering a to be small. The distance between the two surfaces in the radial direction at angle θ is | acos θ| and does not depend on the azimuthal angle. It is seen from the diagram that the surface of the sphere has in effect a surface density σ = σ0 cos θ
Inside any uniformly charged spherical ball, the field is radial and has the magnitude given by Gauss’s theorm
In vector notation, using the fact the V must be measured from the centre of the ball, we get, for the present case
When vector K is the unit vector along the polar axis from which θ is measured.